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Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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%I #39 Apr 28 2020 07:32:32

%S 1,0,1,0,1,2,0,3,10,8,0,3,27,54,31,0,5,70,255,336,147,0,11,223,1222,

%T 2692,2580,899,0,13,508,4467,15512,25330,19566,5777,0,19,1193,15540,

%U 78819,194075,248976,160377,41024,0,27,2822,52981,375440,1303250,2463534,2593339,1430288,322488

%N Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A327244/b327244.txt">Rows n = 0..140, flattened</a>

%F Sum_{k=1..n} k * T(n,k) = A327595(n).

%e T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.

%e T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.

%e T(3,3) = 8: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1, 2;

%e 0, 3, 10, 8;

%e 0, 3, 27, 54, 31;

%e 0, 5, 70, 255, 336, 147;

%e 0, 11, 223, 1222, 2692, 2580, 899;

%e 0, 13, 508, 4467, 15512, 25330, 19566, 5777;

%e 0, 19, 1193, 15540, 78819, 194075, 248976, 160377, 41024;

%e ...

%p C:= binomial:

%p b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(

%p b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1,i),j), j=0..n/i)))

%p end:

%p T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t c = Binomial;

%t b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p+j]/j!*c[c[k + i - 1, i], j], {j, 0, n/i}]]];

%t T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 28 2020, after _Alois P. Heinz_ *)

%Y Columns k=0-2 give: A000007, A032020 (for n>0), A327841.

%Y Main diagonal gives A120774.

%Y Row sums give A309670.

%Y T(2n,n) gives A327596.

%Y Cf. A327245, A327595.

%K nonn,tabl

%O 0,6

%A _Alois P. Heinz_, Sep 14 2019